An approximating approach for boundary control of optimal mixing via Navier-Stokes flows
An approximating approach for boundary control of optimal mixing via Navier-Stokes flows
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DOI：
10.1016/j.jde.2019.06.009
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发表时间：
2019-11
影响因子：
2.4
通讯作者：
Weiwei Hu;Jiahong Wu
中科院分区：
文献类型：
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作者：
Weiwei Hu;Jiahong Wu
本文研究了Ω⊂R2中非耗散标量N-S流的最优混合问题的近似控制设计，目标是在给定的最终时刻T&gT；0通过N-S滑移边界控制实现最优混合，其中H_1(Ω)的对偶空间(H_1(Ω))的索伯列夫范数用于量化混合.我们将研究受输运方程支配的被动标量和主动标量。与我们以前的工作相比，我们目前的方法将导致一个更透明的最优系统来表征最优解[12]。这是通过首先在输运方程中引入一个小的扩散系数，然后建立当扩散系数收敛到零时近似控制问题收敛到原问题的严格分析来实现的。此外，还得到了最优解的唯一性。
The present work focuses on an approximating control design for optimal mixing of a non-dissipative scalar via Navier-Stokes flows in an open bounded and connected domain Ω⊂ R 2. The objective is to achieve optimal mixing at a given final time T> 0, via the active control of the flow velocity through the Navier slip boundary control, where Sobolev norm for the dual space (H 1 (Ω))′ of H 1 (Ω) is adopted for quantifying mixing. Both passive and active scalars governed by the transport equation will be investigated. Our current approach will lead to a more transparent optimality system for characterizing the optimal solution compared to our previous work [12]. This is achieved by first introducing a small diffusivity to the transport equation and then establishing a rigorous analysis of convergence of the approximating control problem to the original one as the diffusivity converges to zero. Moreover, uniqueness of the optimal solution is obtained.